Projects List available now Project proposal (2 pages): 1 st of June Project idea presentation: 8 th of June Optics Final Project presentation: 20 th of July Project report Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Real Lens Optics Outline � Refraction, focusing, formulas � Field of view, sensor format � Aperture and depth of field � Aberrations Acknowledgements for slides � Steve Marschner, Bennett Wilburn, Pat Hanrahan, Marc Levoy Cutaway section of a Vivitar Series 1 90mm f/2.5 lens Cover photo, Kingslake, Optics in Photography Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Pinhole Camera Pinhole camera Large pinhole gives geometric blur Small pinhole gives diffraction blur Optimal pinhole gives very little light � for 35mm format is around f/200 image: Hecht image: Wandell Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Page 1
Diffraction The Reason for Lenses diffraction from a circular aperture: Huygens: every point on a wavefront can be Airy rings considered as a source of spherical wavelets Fresnel: the amplitude of the optical field is the superposition of these waves, considering amplitude and phase Fraunhofer: resulting far-field diffraction pattern images: Hecht 1987 Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Purpose of lens Purpose of lens Produce bright but still sharp image Produce bright but still sharp image Focus rays emerging from a point to a point Focus rays emerging from a point to a point Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Paraxial Refraction Paraxial Refraction “First order” (or Gaussian) optics Refraction governed by Snell’s Law 1. assume e = 0 n sin i = n’ sin i’ 2. assume sin a = tan a ~ a n i ≈ ≈ n’ i’ (Gaussian optics for small angles) ≈ ≈ i i’ a (n) (n’) e Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Page 2
Paraxial Refraction Paraxial Refraction What is z’? i i’ h r i u a i’ h P P' r z z’ u a P P' i = u + a a = u’ + i’ z z’ u = h / z u’ = h / z’ a = h / r n i = n’ i’ Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Paraxial Refraction Focal length i i’ h r r r u a P P' z’ z z’ focal length i = u + a a = u’ + i’ n ( u + a) = n’ ( u’ – a ) z = inf u = h / z u’ = h / z’ n (h/z + h/r) = n’ (h/z’ – h/r) n/r = n’/z’ – n’/r n/z + n/r = n’/z’ – n’/r a = h / r z’ = f = focal length = r/2(n-1) n i = n’ i’ Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Focal Points and Focal Lengths Gauss’ Ray Tracing Construction To focus: move lens relative to backplane Parallel Ray 1 1 1 Focal Ray = + Chief Ray ′ z z f Object Image Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Page 3
Real Image Magnifying Glass Virtual Image Parallel Ray Focal Ray Object Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Thick lenses The “center of perspective” Complex optical system is characterized by a few numbers In a thin lens, the chief ray traverses the lens (through its optical center) without changing direction In a thick lens, the intersections of this ray with the optical axis are called the nodal points For a lens in air, these coincide with the principal points The first nodal point is the center of perspective image: Smith 2000 image: Hecht 1987 Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Focal length and magnification Lens-makers Formula Refractive Power 1 1 1 1 ′ P = ( n − n ) − = = diopters R R f m 1 2 Biconvex Pos. Meniscus Plano concave Plano-convex Biconcave Neg. meniscus Convex = Converging Concave = Diverging image: Kingslake 1992 image: Smith 2000 Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Page 4
Convex and Concave Lenses Focal length and field of view � positive vs. negative focal length Changing the magnification lets us move back from a subject, while maintaining its size on the image Moving back changes perspective relationships From (a) to (c), we’ve moved back from the subject and employed lenses with longer focal lengths image: Kingslake 1992 Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Field of View Field of View images: London and Upton images: London and Upton Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Effects of image format Effects of image format Field of view Smaller formats have... � shorter focal length for same field of view, as fov filmsize = tan we’ve seen 2 2 f � smaller aperture size for same f-number � leads to larger depth of field Types of lenses � lighter, smaller lens for same design � Film camera � enables use of bulkier designs � 36mm x 24mm filmsize Beware: diffraction does not scale down! � 50mm focal length = 40º field of view � smaller apertures suffer more from diffraction � Digital camera � field of view is 2/3 of film for given focal length images: dpreview.com Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Page 5
Aperture: Stops and Pupils Aperture Irradiance on sensor is proportional to � square of aperture diameter A � inverse square of sensor distance (~ focal length) Aperture N therefore specified relative to focal length f N = A � numbers like “f/1.4” – for 50mm lens, aperture is ~35mm � exposure proportional to square of F-number, and independent of actual focal length of lens! • Principal effect: changes exposure Doubling series is traditional for exposure • Side effect: depth of field � therefore the familiar (rounded) sqrt(2) series � 1.4, 2.0, 2.8, 4.0, 5.6, 8.0, 11, 16, 22, 32, … Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 How low can N be? Depth of Field Canon EOS 50mm f/1.0 (discontinued) Principal planes are the paraxial approximation of a spherical “equivalent refracting surface” 1 N = 2 sin θ ' Lowest N (in air) is f/0.5 Lowest N in SLR lenses is f/1.0 images: London and Upton image: Kingslake 1992 Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Depth of focus Depth of Field (in object space) (in image space) tolerance for placing the focus plane the range of depths where the object will be in focus C’ - circle of confusion Note that distance from (in-focus) film plane to front versus back of depth of focus differ www.cambridgeincolour.com image: Kingslake 1992 Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Page 6
Depth of field Numerical Aperture (in object space) total depth of field (i.e. both sides of in-focus plane) NA = θ n sin 2 2 N C U D tot = 2 � The size of the finest detail that can be resolved f is proportional to λ /NA. where (from Goldberg) � larger numerical aperture � � resolve finer detail � � � N = F-number of lens � C = size of circle of confusion (on image) � U = distance to focused plane (in object space) � f = focal length of lens hyperfocal distance � back focal depth becomes infinite when U = f 2 / C N Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Numerical Aperture vs. F-Number Examples 2 2 N C U 1 f /# ≈ D tot = 2 f 2 NA 1 N = f/4, C = 8 � , f /# = ≈ ( 1 − m ) f /# w U = 1m, f = 50mm 2 NA � D tot = 13mm working f-number: f /# w N = f/16, C = 8 � , U = 9mm, f = 65mm distance-related magnification: m � Canon MP-E at 5:1 (macro lens) � use N’ = (1+M)N at short distances (M=5 here) relevant for systems with high magnification (microscopes or marco lenses) � D tot = 0.05mm ! image: Charles Chien Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Tilt and Shift Lens Diffraction Limit Lens shift simply moves the optical axis with regard to Diameter d of 70% radius of the Airy disc the film. f d = 1 . 22 λ � change of perspective (sheared perspective) a Tilt allows for applying Scheimpflug principle � all points on a tilted plane in focus single spot barely resolved no longer resolved image: wikipedia Computational Photography Hendrik Lensch, Summer 2007 Computational Photography Hendrik Lensch, Summer 2007 Page 7
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